The BBC Trust's editorial standards committee (ESC) has apologised over a joke made by Frankie Boyle on Radio 4 comedy show Political Animal in 2008.
One person complained when the Scottish comedian compared the state of Palestine to a cake "being punched to pieces by a very angry Jew".
The BBC's complaints unit responded in December 2008, and agreed the comment was "inappropriate and offensive".
But the complainant was dissatisfied and took the issue to the BBC Trust.
He expressed concern that the remark had gone through the editorial process "without ringing any alarm bells".
The complaint was considered by the ESC, which is the ultimate arbiter in the BBC complaints process.
It has now responded, saying it endorsed the earlier finding about the use of the word "Jew".
It added: "As a result, the committee wished to apologise to the complainant on behalf of the BBC for any offence the remark may have caused him and other listeners to the programme."
The committee noted that the BBC had informed the radio show's production company that the material was unacceptable and breached guidelines.
Radio 4's commissioning editor also discussed the matter with Boyle's agent, who confirmed she had in turn raised the issue with her client.
The committee was satisfied "that the breach, while serious, did not require an apology or correction from the BBC online or on air.
"In conclusion, the committee stated that it was very sorry that the breach of editorial standards had occurred in this case, but was satisfied that no further action was required."
Boyle, 37, found himself in hot water again recently when the mother of a five-year-old girl with Down's syndrome criticised him for poking fun at people with the condition during a live show.
And last year, Mock The Week's producers were criticised by the BBC Trust over comments Boyle made on the show about swimmer Rebecca Adlington's appearance.
Another complaint against Mock The Week, relating to a comment Boyle made about the Queen, was not upheld as a breach of editorial standards by the trust.